A025529 a(n) = (1/1 + 1/2 + ... + 1/n)*lcm{1,2,...,n}.

1, 3, 11, 25, 137, 147, 1089, 2283, 7129, 7381, 83711, 86021, 1145993, 1171733, 1195757, 2436559, 42142223, 42822903, 825887397, 837527025, 848612385, 859193865, 19994251455, 20217344325, 102157567401, 103187226801, 312536252003, 315404588903, 9227046511387

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Comments

First column of A027446. - Eric Desbiaux, Mar 29 2013

From Amiram Eldar and Thomas Ordowski, Aug 07 2019: (Start)

By Wolstenholme's theorem, if p > 3 is a prime, then p^2 | a(p-1).

Conjecture: for n > 3, if n^2 | a(n-1), then n is a prime.

Note that if n = p^2 with prime p > 3, then n | a(n-1).

It seems that composite numbers n such that n | a(n-1) are only the squares n = p^2 of primes p > 3.

Primes p such that p^3 | a(p-1) are the Wolstenholme primes A088164.

The n-th triangular number n(n+1)/2 | a(n) for n = 1, 2, 6, 4422, ... (End)

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• Frank A. Haight, and Robert B. Jones., "A probabilistic treatment of qualitative data with special reference to word association tests." Journal of Mathematical Psychology 11.3 (1974): 237-244. [Denominators of fractions in Eq. 21.] [Annotated scanned copy]
• Frank A. Haight and N. J. A. Sloane, Correspondence, 1975
• Yilmaz Simsek, Derivation of Computational Formulas for certain class of finite sums: Approach to Generating functions arising from p-adic integrals and special functions, arXiv:2108.10756 [math.NT], 2021.

Crossrefs

Differs from A096617 at 7th term.

Cf. A001008, A002805, A027446, A110566.

Programs

GAP

List([1..30],n->Sum([1..n],k->1/k)*Lcm([1..n])); # Muniru A Asiru, Apr 02 2018

Magma

[HarmonicNumber(n)*Lcm([1..n]):n in [1..30]]; // Marius A. Burtea, Aug 07 2019

Maple

a:= n-> add(1/k, k=1..n)*ilcm($1..n):
seq(a(n), n=1..30);  # Alois P. Heinz, Mar 14 2013

Mathematica

Table[HarmonicNumber[n]*LCM @@ Range[n], {n, 27}] (* Arkadiusz Wesolowski, Mar 29 2012 *)

PARI

a(n) = sum(k=1, n, 1/k)*lcm([1..n]); \\ Michel Marcus, Apr 02 2018

Formula

a(n) = A001008(n)*A110566(n). - Arkadiusz Wesolowski, Mar 29 2012

a(n) = Sum_{k=1..n} lcm(1,2,...,n)/k. - Thomas Ordowski, Aug 07 2019